please don’t erase the following for the moment


Models for nn-image factorization

The following gives a sufficient condition for modeling n-image factorizations in some (∞,1)-toposes with particularly convenient presentation.


Let CC be a site with enough points, so that the weak equivalences in sPSh(C) locsPSh(C)_{\mathrm{loc}} are detected on stalks (this prop.). Then given a morphism of Kan complex-valued simplicial presheaves

f:XY f \colon X \longrightarrow Y

such that both XX and YY are homotopy k-types for some finite kk \in \mathbb{N}, then its n-image factorization in the (∞,1)-topos L lwhesPSh(C) locL_{lwhe} sPSh(C)_{loc} for any nn \in \mathbb{N} is presented by any factorization Xim n(f)YX \longrightarrow im_{n}(f) \longrightarrow Y in sPSh(C)sPSh(C) through some Kan-complex valued simplicial presheaf im n(f)im_n(f) such that for each object UCU \in C the simplicial homotopy groups satisfy the following conditions:

  1. π <n(X(U)(im n(f))(U))\pi_{\bullet \lt n}\left(X(U) \to (im_{n}(f))(U)\right) are isomorphisms;

  2. π n(X(U)(im n(f))(U)Y(U))\pi_n\left(X(U) \to (im_{n}(f))(U)\to Y(U)\right) is the (epi,mono) factorization of π n(f(U))\pi_n(f(U));

  3. π >n((im n(f))(U)Y(U))\pi_{\bullet \gt n}\left((im_{n}(f))(U) \to Y(U)\right) are isomorphisms.


Evalutation on stalks is a filtered colimit which preserves the finite limits and finite colimits that go into the definition of simplicial homotopy groups. Therefore the global conditions assumed on the simplicial homotopy groups imply that the same kind of conditions holds for the stalkwise homotopy groups. These are the categorical homotopy groups in L lwhesPSh(C) locL_{lwhe} sPSh(C)_{loc}. By this prop. and this def. we may recognize nn-truncation of morphisms on categorical homotopy groups (using the assumption that XX and YY are kk-truncated for some kk). Therefore the claim now follows from the stalkwise long exact sequence of homotopy groups.

In order to appeal to prop. 1 we are interested in explicit models for nn-image factorization of morphisms of Kan complexes. The following gives such for the special case that the the morphism of Kan complexes is the image under the Dold-Kan correspondence of a chain map between chain complexes.


Let f :V W f_\bullet \colon V_\bullet \longrightarrow W_\bullet be a chain map between chain complexes

For nn \in \mathbb{N}, consider the abelian group

(im n+1(f)) ncoker(ker( V)ker(f n)V n) (im_{n+1}(f))_n \;\coloneqq\; coker(\, ker(\partial_V) \cap ker(f_n) \to V_n \,)

For the following it is helpful to think of this abelian group in the following equivalent ways.

Define an equivalence relation on V nV_n by

(v nv n)(( Vv n= Vv n)and(f n(v n)=f n(v n))). \left( v_n \sim v'_n \right) \;\Leftrightarrow\; \left( (\partial_V v_n = \partial_V v'_n) \;\text{and}\; (f_n(v_n) = f_n(v'_n)) \right) \,.


(im n+1(f)) nV n/ (im_{n+1}(f))_n \simeq V_n/_\sim

is equivalently the set of equivalence classes of this equivalence relation, which inherits abelian group structure since the eqivalence relation is linear.

This is because the equivalence relation says equivalently that

(v nv n)(v nv nker( V)ker(f n)) \left( v_n \sim v'_n \right) \;\Leftrightarrow\; \left( v_n - v'_n \;\in\; ker(\partial_V) \cap ker(f_n) \right)

and hence is generated under linearity by

(v n0)(v nker( V)ker(f n)). \left( v_n \sim 0 \right) \;\Leftrightarrow\; \left( v_n \in ker(\partial_V) \cap ker(f_n) \right) \,.

Moreover, notice that the Dold-Kan correspondence

DK:Ch 0KanCplx DK \;\colon\; Ch_{\bullet \geq 0} \longrightarrow KanCplx

factors through globular strict omega-groupoids (here). An n-morphism in the strict omega-groupoid DK(V )DK(V_\bullet) is of the form

(v n1)AAv nAA(v n1+v n). (v_{n-1}) \overset{\phantom{AA}v_n\phantom{AA}}{\longrightarrow} (v_{n-1} + \partial v_n) \,.

In terms of these morphisms the equivalence relation above says that two of them are equivalent precisely if

  1. they are “parallel morphisms” in that they have the same source and target;

  2. they have the same image under ff in the n-morphisms of DK(W )DK(W_\bullet).

This suggests yet another equivalent way to think of (im n+1(f)) n(im_{n+1}(f))_n: it is the disjoint union over the target (n1)(n-1)-cells in V n1V_{n-1} of the images under ff of the sets of nn-cells from zero to that target:

(im n+1(f)) nv n1V n1{f n(v n)|v nV nandv n=v n1}. (im_{n+1}(f))_n \simeq \underset{v_{n-1} \in V_{n-1}}{\sqcup} \left\{ f_n(v_n) \vert v_n \in V_n \,\text{and}\,\partial v_n = v_{n-1} \right\} \,.

Let f :V W f_\bullet \colon V_\bullet \longrightarrow W_\bullet be a chain map between chain complexes and let nn \in \mathbb{N}. Recall the abelian group v n1{f n(v n)|v n=v n1}\underset{v_{n-1}}{\sqcup}\{f_n(v_n) \vert \partial v_n = v_{n-1}\} from remark 1.

The following diagram of abelian groups commutes:

V W W V n+3 f n+3 W n+3 = W n+3 V W W V n+2 f n+2 W n+2 = W n+2 V W W V n+1 f n+1 {w n+1|v n: Ww n+1=f n(v n), Vv n=0,} W n+1 V W W V n (f n, V) v n1{f n(v n)| Vv n=v n1} W n V (f n(v n), Vv n) Vv n W V n1 = V n1 f n1 W n1 V V W V n2 = V n2 f n2 W n2 V V W \array{ \vdots && \vdots && \vdots \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{W}}} && \downarrow^{\mathrlap{\partial_{W}}} \\ V_{n+3} &\overset{f_{n+3}}{\longrightarrow}& W_{n+3} &\overset{=}{\longrightarrow}& W_{n+3} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{W}}} && \downarrow^{\mathrlap{\partial_{W}}} \\ V_{n+2} &\overset{f_{n+2}}{\longrightarrow}& W_{n+2} &\overset{=}{\longrightarrow}& W_{n+2} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{ \partial_W } } && \downarrow^{\mathrlap{\partial_{W}}} \\ V_{n+1} &\overset{f_{n+1}}{\longrightarrow}& \left\{ w_{n+1} | \exists v_n : \partial_W w_{n+1} = f_n(v_n), \partial_V v_n = 0, \right\} &\overset{}{\longrightarrow}& W_{n+1} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\partial_W} && \downarrow^{\mathrlap{\partial_{W}}} \\ V_n &\overset{ (f_n, \partial_V) }{\longrightarrow}& \underset{v_{n-1}}{\sqcup} \left\{ f_n(v_n) \vert \partial_V v_n = v_{n-1} \right\} &\overset{ }{\longrightarrow}& W_n \\ \downarrow^{\mathrlap{\partial_V}} && \downarrow^{\mathrlap{(f_n(v_n),\partial_V v_n) \mapsto \partial_V v_n}} && \downarrow^{\mathrlap{\partial_W}} \\ V_{n-1} &\overset{=}{\longrightarrow}& V_{n-1} &\overset{f_{n-1}}{\longrightarrow}& W_{n-1} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_W}} \\ V_{n-2} &\overset{=}{\longrightarrow}& V_{n-2} &\overset{f_{n-2}}{\longrightarrow}& W_{n-2} \\ \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_W}} \\ \vdots && \vdots && \vdots }

Moreover, the middle vertical sequence is a chain complex im n+1(f) im_{n+1}(f)_\bullet, and hence the diagram gives a factorization of f f_\bullet into two chain maps

f :V im n+1(f) W . f_\bullet \;\colon\; V_\bullet \longrightarrow im_{n+1}(f)_\bullet \longrightarrow W_\bullet \,.

Finally, this is a model for the (n+1)-image factorization of ff in that on homology groups the following holds:

  1. H <n(V)H <n(im n+1(f))H_{\bullet \lt n}(V) \overset{\simeq}{\to} H_{\bullet \lt n}(im_{n+1}(f)) are isomorphisms;

  2. H n(V)H n(im n+1(f))H n(W)H_n(V) \to H_n(im_{n+1}(f)) \hookrightarrow H_n(W) is the image factorization of H n(f)H_n(f);

  3. H >n(im n+1(f))H >n(W)H_{\bullet \gt n}(im_{n+1}(f)) \overset{\simeq}{\to} H_{\bullet \gt n}(W) are isomorphisms.

Proof (but check)

This follows by elementary and straightforward direct inspection.

Moduli of circle nn-connections


For pp \in \mathbb{N} and kp+1k \leq p+1 write

B p+1U(1) conn kDK(U(1)Ω 1Ω 2Ω k00)sPSh(CartSp) \mathbf{B}^{p+1}U(1)_{conn^k} \coloneqq DK \left( U(1) \to \Omega^1 \to \Omega^2 \to \cdots \to \Omega^k \to 0 \to \cdots \to 0 \right) \;\in\; sPSh(CartSp)

for the simplicial presheaf which is the image under the Dold-Kan correspondence of the presheaf of chain complexes which is the Deligne complex starting with the presheaf represented by U(1)U(1) in degree p+1p+1 and truncated to the differential kk-forms, as shown.

Since the DKDK map sends surjections of chain complexes to Kan fibrations, the canonical projection maps yield a tower of objectwise Kan fibrations of the following form:

B p+1U(1) conn=B p+1U(1) conn p+1B p+1U(1) conn pB p+1U(1) conn p1B p+1U(1) conn 1B p+1U(1) conn 0=B p+1U(1). \mathbf{B}^{p+1}U(1)_{conn} = \mathbf{B}^{p+1}U(1)_{conn^{p+1}} \longrightarrow \mathbf{B}^{p+1}U(1)_{conn^{p}} \longrightarrow \mathbf{B}^{p+1}U(1)_{conn^{p-1}} \longrightarrow \mathbf{B}^{p+1}U(1)_{conn^1} \longrightarrow \mathbf{B}^{p+1}U(1)_{conn^0} = \mathbf{B}^{p+1}U(1) \,.

For Σ\Sigma a smooth manifold, write

(B pU(1))Conn(Σ)sPSh(CartSp) (\mathbf{B}^p U(1)) \mathbf{Conn}(\Sigma) \in sPSh(CartSp)

for the image under the Dold-Kan correspondence of the presheaf of chain complexes which to UCartSpU \in CartSp assigns the vertical Cech-Deligne complex on Σ×UU\Sigma \times U \to U in the given degree, i.e. the Cech-Deligne complex involving differential forms on Σ×U\Sigma \times U that have no leg along UU, i.e. those in Ω ,0(Σ×U)\Omega^{\bullet,0}(\Sigma \times U).

Differential concretification on contractibles

We first consider differential concretification on geometrically contractible base spaces. Once this is established, then the general differential concretification follows simply by stackifying along the base space.


(differential concretification for higher circle connections on contractibles)

Let Σ\Sigma be a contractible smooth manifold. For pp \in \mathbb{N} write

(B pU(1))Conn 0(Σ)[Σ,B p+1U(1)] (\mathbf{B}^p U(1))\mathbf{Conn}_0(\Sigma) \coloneqq [\Sigma, \mathbf{B}^{p+1}U(1)]

and then for 0kp0 \leq k \leq p define inductively

(B pU(1))Conn k+1(Σ)im p+1k([Σ,B(B pU(1)) conn k+1][Σ,B(B pU(1)) conn k+1]× h[Σ,B(B pU(1)) conn k](B pU(1))Conn k(Σ)). (\mathbf{B}^p U(1))\mathbf{Conn}_{k+1}(\Sigma) \coloneqq im_{p+1-k} \left( [\Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn^{k+1}}] \longrightarrow \sharp [ \Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn^{k+1}} ] \underset{\sharp[\Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn^k}]}{\times^h} (\mathbf{B}^p U(1))\mathbf{Conn}_k(\Sigma) \right) \,.

Let Σ\Sigma be a contractible smooth manifold. Then there is a weak equivalence

(B pU(1))Conn p+1(Σ)(B pU(1))Conn(Σ), (\mathbf{B}^p U(1)) \mathbf{Conn}_{p+1}(\Sigma) \simeq (\mathbf{B}^p U(1)) \mathbf{Conn}(\Sigma) \,,

from the inductively defined object from def. 3 to the moduli object from def. 2.


By the assumption that Σ\Sigma is contractible, the Cech-direction of the Cech-Deligne double complex is trivial and so we have for all UCartSpU \in CartSp and 0kp0 \leq k \leq p weak equivalences of the form

[Σ,B p+1U(1) conn k](U)DK(C (Σ×U,U(1))Ω 1(Σ×U)Ω 2(Σ×U)Ω p+1(Σ×U)) [\Sigma, \mathbf{B}^{p+1}U(1)_{conn^k}](U) \;\simeq\; DK\left( C^\infty(\Sigma \times U, U(1)) \to \Omega^1(\Sigma \times U) \to \Omega^2(\Sigma \times U) \to \cdots \to \Omega^{p+1}(\Sigma \times U) \right)


(B pU(1))Conn(Σ)DK(C (Σ×U,U(1))Ω 1,0(Σ×U)Ω 2,0(Σ×U)Ω p+1,0(Σ×U)). (\mathbf{B}^p U(1))\mathbf{Conn}(\Sigma) \simeq DK\left( C^\infty(\Sigma \times U, U(1)) \to \Omega^{1,0}(\Sigma \times U) \to \Omega^{2,0}(\Sigma \times U) \to \cdots \to \Omega^{p+1,0}(\Sigma \times U) \right) \,.

We claim now for all kpk \leq p that

(B pU(1))Conn k(Σ)DK(C (Σ×U,U(1))Ω 1,0(Σ×U)Ω k,0(Σ×U)00). (\mathbf{B}^p U(1))\mathbf{Conn}_k(\Sigma) \simeq DK\left( C^\infty(\Sigma \times U, U(1)) \to \Omega^{1,0}(\Sigma \times U) \to \cdots \to \Omega^{k,0}(\Sigma \times U) \to 0 \to \cdots \to 0 \right) \,.

For k=pk = p this is the statement to be shown. Hence we may now prove this by induction.

It is manifestly true for k=0k = 0. Hence suppose it is true for some k<pk \lt p. Observe then that

[Σ,B p+1U(1) conn k+1][Σ,B p+1U(1) conn k] \sharp [\Sigma, \mathbf{B}^{p+1}U(1)_{conn^{k+1}}] \longrightarrow \sharp [\Sigma, \mathbf{B}^{p+1}U(1)_{conn^k}]

is an objectwise Kan fibration, because so is B p+1U(1) conn k+1B p+1U(1) conn k\mathbf{B}^{p+1}U(1)_{conn^{k+1}} \to \mathbf{B}^{p+1}U(1)_{conn^k} by def. 1, and both [Σ,][\Sigma,-] as well as \sharp are right Quillen functors from sPSh(C)sPSh(C) with its global projective model structre to itself.

It follows (this prop.) that the homotopy fiber product in question is presented by the plain 1-categorical fiber product. Since DKDK is right adjoint, this in turn is given by the degreewise fiber product of the corresponding chain complexes. By direct inspection this means that

[Σ,B(B pU(1)) conn k+1]× h[Σ,B(B pU(1)) conn k](B pU(1))Conn k(Σ) DK(C (Σ×U,U(1))Ω 1,0(Σ×U)Ω k,0(Σ×U)(Ω k+1(Σ×))(U)00) \begin{aligned} & \sharp [ \Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn_{k+1}} ] \underset{\sharp[\Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn_k}]}{\times^h} (\mathbf{B}^p U(1))\mathbf{Conn}_k(\Sigma) \\ & \simeq DK \left( C^\infty(\Sigma \times U, U(1)) \to \Omega^{1,0}(\Sigma \times U) \to \cdots \to \Omega^{k,0}(\Sigma \times U) \to (\sharp \Omega^{k+1}(\Sigma \times -))(U) \to 0 \to \cdots \to 0 \right) \end{aligned}

Hence we are now reduced to computing the (p+1k)(p+1-k) image of

DK(C (Σ×U) Ω 1(Σ×U) Ω k(Σ×U) Ω k+1(Σ×U) 0 0) DK(C (Σ×U,U(1)) Ω 1,0(Σ×U) Ω k,0(Σ×U) (Ω k+1(Σ×))(U) 0 0) \array{ DK ( C^\infty(\Sigma \times U) &\to& \Omega^1(\Sigma \times U) &\to& \cdots &\to& \Omega^{k}(\Sigma \times U) &\to& \Omega^{k+1}(\Sigma \times U) &\to& 0 &\to& \cdots &\to& 0 ) \\ \downarrow && \downarrow && && \downarrow && \downarrow && \downarrow && && \downarrow \\ DK ( C^\infty(\Sigma \times U, U(1)) &\to& \Omega^{1,0}(\Sigma \times U) &\to& \cdots &\to& \Omega^{k,0}(\Sigma \times U) &\to& (\sharp \Omega^{k+1}(\Sigma \times -))(U) &\to& 0 &\to& \cdots &\to& 0 ) }

Observe that in degree (p+1)(k+1)(p+1)-(k+1) the ordinary image is

im(Ω k+1(Σ×U)(Ω k+1(Σ×))(U))Ω k+1,0(Σ×U) im\left( \Omega^{k+1}(\Sigma \times U) \to (\sharp \Omega^{k+1}(\Sigma \times -))(U) \right) \simeq \Omega^{k+1,0}(\Sigma \times U)

With this the induction follows by prop. 1 and prop. \ref{nImageFactorizationOnChainComplex}.

General differential concretification


(differential concretification of moduli for higher connection)

For Σ\Sigma a smooth manifold, define for pp \in \mathbb{N}

(B pU(1))Conn p+1(Σ)lim i h(B pU(1))Conn p+1(U i) (\mathbf{B}^{p}U(1)) \mathbf{Conn}_{p+1}(\Sigma) \;\simeq\; \underset{\longleftarrow}{\lim}^h_i \; (\mathbf{B}^p U(1)) \mathbf{Conn}_{p+1}(U_i)

to be the homotopy limit over the differential concretifications from def. 3 of contractibles U iU_i, for

Σlim i hU i \Sigma \simeq \underset{\longrightarrow}{\lim}_i^h U_i

a presentation of Σ\Sigma as a homotopy colimit of contractible manifolds (e.g. the realization of the Cech nerve of a good open cover).


For Σ\Sigma a smooth manifold, then the differential concretifiction of def. 4 is equivalent to the moduli object from def. 2:

(B pU(1))Conn p+1(Σ)(B pU(1))Conn(Σ). (\mathbf{B}^p U(1)) \mathbf{Conn}_{p+1}(\Sigma) \simeq (\mathbf{B}^{p}U(1)) \mathbf{Conn}(\Sigma) \,.

Let Σlim i hU i\Sigma \simeq \underset{\longrightarrow}{\lim}_i^h U_i be the realization of the Cech nerve of a good open cover. Then

lim i(B pU(1))Conn p+1(U i) \underset{\longleftarrow}{\lim}_i (\mathbf{B}^p U(1))\mathbf{Conn}_{p+1}(U_i)

is equivalently the image under DK of the corresponding Cech hypercomplex with coefficients in the presheaf of chain complexes (B pU(1))Conn p+1()(\mathbf{B}^p U(1))\mathbf{Conn}_{p+1}(-). By lemma 1 this is the vertical Deligne complex, and hence the claim follows.

Revised on March 20, 2017 11:24:11 by Urs Schreiber (